Matches in SemOpenAlex for { <https://semopenalex.org/work/W2937263345> ?p ?o ?g. }
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- W2937263345 abstract "Abstract The main purpose of this paper is to prove several sharp singular Trudinger–Moser-type inequalities on domains in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> {mathbb{R}^{N}} with infinite volume on the Sobolev-type spaces <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>D</m:mi> <m:mrow> <m:mi>N</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {D^{N,q}(mathbb{R}^{N})} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>q</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {qgeq 1} , the completion of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msubsup> <m:mi>C</m:mi> <m:mn>0</m:mn> <m:mi mathvariant=normal>∞</m:mi> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {C_{0}^{infty}(mathbb{R}^{N})} under the norm <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>∥</m:mo> </m:mrow> <m:mi>N</m:mi> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mi>u</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:mi>q</m:mi> </m:msub> </m:mrow> </m:math> {|nabla u|_{N}+|u|_{q}} . The case <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>q</m:mi> <m:mo>=</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> {q=N} (i.e., <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msup> <m:mi>D</m:mi> <m:mrow> <m:mi>N</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {D^{N,q}(mathbb{R}^{N})=W^{1,N}(mathbb{R}^{N})} ) has been well studied to date. Our goal is to investigate which type of Trudinger–Moser inequality holds under different norms when q changes. We will study these inequalities under two types of constraint: semi-norm type <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>∥</m:mo> </m:mrow> <m:mi>N</m:mi> </m:msub> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {|nabla u|_{N}leq 1} and full-norm type <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msubsup> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>∥</m:mo> </m:mrow> <m:mi>N</m:mi> <m:mi>a</m:mi> </m:msubsup> <m:mo>+</m:mo> <m:msubsup> <m:mrow> <m:mo>∥</m:mo> <m:mi>u</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:mi>q</m:mi> <m:mi>b</m:mi> </m:msubsup> </m:mrow> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {|nabla u|_{N}^{a}+|u|_{q}^{b}leq 1} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>a</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {a>0} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>b</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {b>0} . We will show that the Trudinger–Moser-type inequalities hold if and only if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>b</m:mi> <m:mo>≤</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> {bleq N} . Moreover, the relationship between these inequalities under these two types of constraints will also be investigated. Furthermore, we will also provide versions of exponential type inequalities with exact growth when <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>b</m:mi> <m:mo>></m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> {b>N} ." @default.
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- W2937263345 date "2019-04-12" @default.
- W2937263345 modified "2023-10-11" @default.
- W2937263345 title "Sharp Singular Trudinger–Moser Inequalities Under Different Norms" @default.
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- W2937263345 doi "https://doi.org/10.1515/ans-2019-2042" @default.
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